Stellation

The word stellation comes from the Latin word for star: stella.
Stellation is a process that allows us to derive a new polyhedron from
an existing one by extending the faces until they re-intersect.

Stellating in 2 Dimensions

I think you will find the stellation of polyhedrons much easier to understand
after seeing the process of stellation applied to polygons. So, to that end,
we start with the regular Pentagon and extend the edges until they meet.

The result is the familiar 5-pointed star (Pentagram). You can extend the
edges further, but they never meet again, so apparently there is only one
stellation of the Pentagon.

As a second example, let's try stellating the Octagon. Extending the edges
until they first intersect produces an eight-pointed star shape. This is the
first stellation of the Octagon.

Note, however, this shape can also be interpreted as a Compound of
two equal squares, one rotated 45 degrees from the other. Some of the stellations
of polyhedra that we will look at also have dual interpretations.

If you extend the edges of this new shape, you get another eight-pointed star
shape. This is the second stellation of the Octagon. The appropriateness of
the term Stellation is becoming clear!

Here are a couple of exercises for you: First, convince yourself that
the Triangle and Square have no stellations, that the Pentagon and Hexagon
have one stellation, and that the Heptagon and Octagon have two stellations.
Second, see if you can come up with a formula for how many stellations a
regular N-gon has.

Stellating in 3 Dimensions

We are now ready to turn our attention to the stellation of polyhedra. We
begin with the simplest polyhedron that has a stellated form, the Octahedron.

Consider one of the faces in the in the image at the right. If the three
sides around it are extended, they meet to form a triangluar pyramid.
Repeating this process over every face results in the Stellated Octahedron.

Notice that this shape has a duel interpretation, like we saw with the first
stellation of the Octagon. It is also a compound to two Tetrahedrons!

Consider now the other Platonic Solids.
You will find that the Tetrahedron and Cube have no stellated forms; if you
extend their faces they never re-intersect.

The remaining two have several stellated forms each. The Dodecahedron has
three, and it turns out that they are quite special
(see The Kepler-Poinsot Polyhedra),
and the Icosahedron has an incredible 59 stellated forms!

We can also stellate the Archimedian solids, but little is known about
most of these. It is here that we unexpectedly meet the boundary of
contempory mathematics!